
Many researchers consider large sample properties against a fixed treatment effect independent of the sample size.
From one point of view, a treatment effect is a population quantity which does not change with sample size. But if one develops an asymptotic approximation to the testing environment using a fixed treatment effect, the statistical experiment stops at the first interim analysis with probability one for any consistent test; test statistics degenerate to a point mass; see Section 7.4 in [1]}. Under a fixed treatment effect, the power converges to one and cannot be used to compare different testing procedures. This issue triggered development of various descriptions of asymptotic relative efficiency. The most popular approach is Pitman asymptotic relative efficiency [2]}, where asymptotic power is evaluated under local alternatives; see [3]}. In addition, local alternatives clearly reflect actual practice for experiment planning. Experiments are never planned for a statistical power = 1. Small sample size studies (pre-clinical, animal studies) are planned to detect large effect sizes, moderate sample sizes (typical phase 3 studies) are used to detect moderate effect sizes, and large sample sizes (epidemiological studies, like vaccine studies) are used to detect small differences.
[4]} explored MLEs conditional on stopping, but assumed asymptotic normality to evaluate their uncertainty. [5]} relied on asymptotic normality for evaluating regression coefficients under the Fine–Gray model in GSD settings. [6]} evaluated asymptotic properties with SSR under a fixed alternative for negative binomial random variables.

Some researchers investigate local asymptotic properties when early stopping is not possible or is ancillary to the treatment effect. This also leads to asymptotic normality. [7]} show that without possibility of early stopping “time-sequential joint distributions of many statistics \(\ldots \)  are multivariate normal with an independent increments covariance structure” under local alternatives. These results are generally consistent with the classical results on local asymptotic normality of [8]}, where both mean and variance of the limiting normal distribution depend on the parameter of interest; see Chapter 7 of [9]}. However, Section REF  shows that the possibility of early stopping destroys local asymptotic normality: the limiting distribution of standardized test statistics is a mixture of truncated normal distributions. Similar findings were previously proved for non-ancillary sample size recalculations; see [10]}.

